Synthetic Division
***Only use with linear divisors***
***Use “0” for missing terms****
Uses other than division:
1) To test factors or roots/zeros/solutions:
Synthetic division for a value of “x” with a remainder = 0
indicates that it IS a factor/root/zero/solution.
2) To find missing
factors/roots/zeros/solutions:
Divide by known or given roots in order to factor or solve
remaining polynomial.
3) Remainder Theorem:
INSTEAD of substituting a value of “x” into the polynomial,
the REMAINDER = VALUE.
DIVISION OF POLYNOMIALS
Synthetic Division
***Only use with linear divisors***
***Use “0” for missing terms****
***Value of “x” goes in “box”***
***Bring down 1st coefficient, multiply, add***
***Each division reduces degree by 1***
***Numbers represent new coefficients***
Long Division
***Use with ANY divisor***
***Use “0x” for any missing terms***
***Same process as for numbers: Divide,
Multiply, Subtract, Bring down***
Finding Polynomials
***Use roots/zeros to write factors***
***Always have OPPOSITE of irrational and
imaginary roots too***
***Multiply factors and combine like terms***
***Write in standard form***
Binomial Expansion
***use instead of multiplying binomial factor by
itself repeatedly***
***Use row of numbers from Pascal’s Triangle
with descending powers of first term and
ascending powers of last term***
Graphing Polynomials
End Behavior
***Describes the direction of the right and left
ends of a polynomial function***
***Depends of if degree is EVEN (both same
direction) or ODD (opposite directions)
and
sign of “a” (+ right side up,
-- right side down)***
Roots/ Zeros
***Real roots/zeros represent X-INTERCEPTS***
Solving/Finding roots or zeros
By factoring:
***Check for and factor GCF***
***Technique depends on number of terms***
2 terms: formula
sum of cubes : (𝒂 + 𝒃)(𝒂𝟐 − 𝒂𝒃 + 𝒃𝟐 )
difference of cubes: (𝒂 − 𝒃)(𝒂𝟐 + 𝒂𝒃 + 𝒃𝟐 )
difference of squares: (𝒂 + 𝒃)(𝒂 − 𝒃)
***no sum of squares***
3 terms: guess and check
4 terms: grouping
***set each factor = 0 and solve***
Rational Root Theorem
***use when you can’t factor***
***possible roots =
±𝒇𝒂𝒄𝒕𝒐𝒓𝒔 𝒐𝒇 𝒕𝒉𝒆 𝒍𝒂𝒔𝒕 𝒏𝒖𝒎𝒃𝒆𝒓
±𝒇𝒂𝒄𝒕𝒐𝒓𝒔 𝒐𝒇 "𝒂"
***divide by roots found in calculator until
degree= 1 or 2***
***solve remaining polynomial***
Other Uses of Synthetic Division
1. Is (x-3) a factor of the polynomial 𝟐𝒙𝟑 + 𝟒𝒙𝟐 − 𝟏𝟎𝒙 − 𝟗?
(Show work to justify your answer.) Explain how you can tell.
2. Use the remainder theorem to evaluate
𝟏
𝟐𝒙𝟑 − 𝟒𝒙𝟐 + 𝟏𝟎𝒙 + 𝟓 if x =
𝟐
3. If x = 2 is a root of 𝒙𝟑 − 𝟕𝒙𝟐 + 𝟏𝟒𝒙 − 𝟖 , write the
complete factorization of the polynomial.
4. Is 4i a zero of 𝒙𝟒 − 𝟑𝒙𝟑 + 𝟔𝒙𝟐 − 𝟒𝟖𝒙 − 𝟏𝟔𝟎? (Show work
to justify your answer.)
Finding Polynomials
For each of the following, find the polynomial in standard form:
𝟏
𝟑
𝟑
𝟐
1. roots are x = 3, , and −
2. roots are x = 0 multiplicity of 2, and x = -1 multiplicty 2
3. roots are x = √𝟑, 4
4. root is x = -2i +3
5. (𝒙 + 𝟐)𝟐 − 𝟒(𝒙 − 𝒙𝟒 ) + 𝟏𝟏
6. Use binomial expansion: (𝒙 + 𝟐)𝟓
Solving/Finding Zeros/Roots
Solve each equation according to directions:
1. 3𝑥 3 − 81 = 0 by factoring
2. 𝑥 4 − 18𝑥 2 + 32 = 0 by factoring
3. 𝑥 3 + 6𝑥 2 − 5𝑥 − 30 by factoring
4. 𝑥 4 − 9𝑥 3 + 11𝑥 2 − 19𝑥 − 40 = 0
5. If 𝑥 = 3 − 4𝑖 is a zero of 𝑥 4 − 6𝑥 3 + 29𝑥 2 −
24𝑥 + 100, find all other zeros.
Khan Academy Practices:
“Graphs of Polynomials”- Use roots and end
behavior to match graphs to polynomial
equations.
“Dividing Polynomials with remainders”Use synthetic or long division to divide.
Remainders should be entered as a fraction with
+/- sign in front.